Question

Find a polynomial equation of minimum degree with rational coefficients, having `sqrt(5) - sqrt(3)` as a root

Answer 1

The given one roots of the polynomial equation are `(sqrt(5) - sqrt(3))`

The other roots are `(sqrt(5) + sqrt(3), (- sqrt(5) + sqrt(3))` and `(- sqrt(5) - sqrt(3))`.

The quadratic factor with roots `(sqrt(5) - sqrt(3))` and `(sqrt(5) + sqrt(3))` is

= x² – x(S.O.R) + P.O.R

= `x^2 - x(2sqrt(5)) + (sqrt(5) - sqrt(3))(sqrt(5) + sqrt(3))`

= `x^2 - 2sqrt(5)x + 2`

The other quadratic factors with roots `(- sqrt(5) + sqrt(3))(- sqrt(5) - sqrt(3))` is

= x² – x (S.O.R) + P.O.R

= `x^2 - x(- 2sqrt(5)) + (5 - 3)`

= `x^2 + 2sqrt(5)x + 2`

To rationalize the co-efficients with minimum degree

`(x^2 - 2sqrt(5)x + 2)(x^2 + 2sqrt(5)x + 2)` = 0

⇒ `(x^2 + 2)^2 - (2sqrt(5)x)^2` = 0

⇒ x⁴ + 4 + 4x² – 20x² = 0

⇒ x⁴ – 16x² + 4 = 0